Étale morphism of varieties over $k$, does bijective on $k$ points imply isomorphism?

Question

I came across this statement in a context where it isn't clear when $k$ is algebraically closed or not, so my question is:

Let $X$ and $Y$ be two varieties over an arbitrary field $k$ and $f: X \to Y$ an étale morphism. Is it true that $f$ is bijective on $k$ points implies $f$ is an isomorphism. I might buy this for $k=\bar{k}$ but in general I'm not sure.

Answer 1

No. For instance, let $k=\mathbb{R}$, let $Y=\operatorname{Spec}(\mathbb{R})$, and let $X=\operatorname{Spec}(\mathbb{R}\times\mathbb{C})$. Then the obvious morphism $f:X\to Y$ is étale and bijective on $\mathbb{R}$-points, but not an isomorphism.